Verify Rolles theorem for the function `f(x)=(x-a)^m(x-b)^n` on the interval `[a , b],` where `m , n` are positive integers.

Explanation

To verify Rolle's theorem for the function \( f(x) = (x-a)^m (x-b)^n \) on the interval \([a, b]\), where \( m \) and \( n \) are positive integers, we will follow these steps:

### Step 1: Check the conditions of Rolle's Theorem
Rolle's theorem states that if a function \( f \) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in \((a, b)\) such that \( f'(c) = 0 \).

### Step 2: Verify continuity and differentiability
The function \( f(x) = (x-a)^m (x-b)^n \) is a polynomial function, and polynomial functions are continuous and differentiable everywhere on the real line. Therefore, \( f(x) \) is continuous on \([a, b]\) and differentiable on \((a, b)\).

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